So with a explosives build, which would be better to back it up with? Melee, or unarmed?

I'd say it's six of one/half a dozen of the other. It only depends on your personal preference. Though there are some pretty deadly Unarmed weapons in the game, maybe more so than Melee. You might want to open up the GECK and have a look and compare.

I personally went with unarmed, since range is almost not a factor between the two (but there is a slight advantage), unarmed seems to strike faster imo plus you don't really need to wait as long to switch from say a grenade to your fist, iirc. As Nicho said, it's all personal preference. I made a bunch of 2-weapon builds, and melee for me went with my sniper build which used almost all scoped-high accuracy rifles (versus assault rifles) and then melee...all stealth.

With the unarmed/explosive build, I'm not worried about stealth as much, just make sure my AR (or DR?) is high enough so that after I lob a few grenades, I can withstand damage and run in and use my fists quickly to finish stuff off.

I use melee, because I always got the feeling that unarmed relies more on LUCK than melee (and explosives don't need that stat). With J Sawyer's mod I didn't use any melee related perks though, because there isn't enough space to fit those in. You should feel free to vary your choices and try to come up with what suits your playstyle best.

I also always had a minigun with me.

I think that melee works well with explosives, as critical are useless for both; on this build I'll have a luck of one, so melee and explosives are the way to go... I think unarmed and energy go well together, better with high crit chance

I went unarmed. I gave my character the story of being a former member of the Brotherhood of Steel, so she would have known a bit about punching things.